Centers for the Kukles homogeneous systems with even degree
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For the polynomial differential system x˙=−y, y˙=x+Qn(x,y), where Qn(x,y) is a homogeneous polynomial of degree n there are the following two conjectures done in 1999. (1) Is it true that the previous system for n≥2 has a center at the origin if and only if its vector field is symmetric about one of
the coordinate axes? (2) Is it true that the origin is an isochronous center of the previous system with the exception of the linear center only if the system has even degree? We give a step forward in the direction of proving both conjectures for all n even. More precisely, we prove both conjectures in the case n=4 and for n≥6 even under the assumption that if the system has a center or an isochronous center at the origin, then it is symmetric with respect to one of the coordinate axes, or it has a local analytic first integral which is continuous in the parameters of the system in a neighborhood of zero in the parameters space. The case of n odd was studied in .
Is part ofJournal Of Applied Analysis And Computation, 2017, vol. 7, núm. 4, p. 1534-1548
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Giné, Jaume; Llibre, Jaume; Valls, Claudia (London Mathematical Society, 2015)For the polynomial differential system x˙ = −y, y˙ = x+Qn(x; y), where Qn(x; y) is a homogeneous polynomial of degree n there are the following two conjectures done in 1999. (1) Is it true that the previous system for ...
Giné, Jaume; Valls, Claudia (Elsevier, 2016)Using tools of computer algebra based on the Gröbner basis theory we derive conditions for the existence of a center on a local center manifold for fifteen seven-parameter families of quadratic systems on R 3. To obtain ...
Giné, Jaume; Llibre, Jaume; Valls, Claudia (Elsevier, 2015)In this paper we classify the centers and the isochronous centers of certain polynomial differential systems in R2 of degree d≥5 odd that in complex notation are ż=(λ+i)z(zz̄)d−52(Az5+Bz4z̄+Cz3z̄2+Dz2z̄3+Ezz̄4+Fz̄5), ...